Maximal-degeneration Gromov–Hausdorff limit conjecture for Calabi–Yau manifolds

Let ${{\cal X}}_{mer}$ be a maximally degenerate Calabi–Yau family at $q=0$, let $X_q^{new}$ be the rescaled family, and let $n$ be the complex dimension. Maximal-degeneration limit conjecture. If ${{\cal X}}_{mer}$ has maximal degeneration at $q=0$, then there is a Gromov–Hausdorff limit $(\overline{Y},g_{\overline{Y}})$ of $X_q^{new}$ such that: (a) it is a compact metric space containing a smooth oriented Riemannian $n$-manifold $(Y,g_Y)$ as a dense open metric subspace, with $Y^{sing}=\overline{Y}\setminus Y$ of Hausdorff dimension at most $n-2$; (b) $Y$ carries an integral affine structure, meaning a torsion-free flat connection $\nabla$ with holonomy in $SL(n,{{\bf Z}})$; (c) $g_Y$ locally has a potential $K$, with $g_{ij}=\partial^2K/\partial x_i\partial x_j$ in affine coordinates; and (d) $\det(g_{ij})=\det(\partial^2K/\partial x_i\partial x_j)=\operatorname{const}$ in affine coordinates. This conjecture describes the expected real Monge–Ampère geometry of maximal Calabi–Yau degenerations; the source gives no resolution.